Spaces of initial conditions for quartic Hamiltonian systems of Painlevé and quasi-Painlevé type

TL;DR

This work extends the geometric Okamoto framework to a broad class of non-autonomous quartic Hamiltonian systems by constructing defining manifolds through cascades of blow-ups and analysing base-point coalescence and coefficient degenerations. The authors classify quasi-Painlevé systems by the resulting Newton polygons and surface types, detailing how movable algebraic poles of various orders arise and linking many subcases to known Painlevé equations (I–IV, and modified PV/PII) via explicit coordinate changes. The approach yields a rich taxonomy of quartic (and cubic) Hamiltonians, revealing both genuine quasi-Painlevé cases with mixed pole structures and Painlevé-type reductions embedded in the quartic family; in particular, Painlevé I–V surfaces appear in several sub-cases, with PV and its relatives emerging in more intricate quartic configurations. Overall, the paper provides a systematic, geometry-driven pathway to identify, regularise, and connect quasi-Painlevé Hamiltonians to classical Painlevé dynamics, offering a foundation for higher-degree generalisations and deeper algebro-geometric insights into the space of initial conditions for complex differential systems.

Abstract

The geometric approach for Painlevé and quasi-Painlevé differential equations in the complex plane is applied to non-autonomous Hamiltonian systems, quartic in the dependent variables. By computing their defining manifolds (analogue of the Okamoto's space of initial conditions in the quasi-Painlevé case), we provide a classification of such systems. We distinguish the various cases by the local behaviour at the movable singularities of the solutions, which are algebraic poles or ordinary poles. The principal cases are categorised by the initial base points of the system in the extended phase space $\mathbb{CP}^2$ and their multiplicities, arising from the coalescence of $4$ simple base points in the generic case. Through the mechanisms of coalescence of base points and degeneration (by setting certain coefficient functions in the Hamiltonian to $0$), all possible sub-cases of quartic Hamiltonian systems with the quasi-Painlevé property are obtained, and are characterised by their corresponding Newton polygons. As particular sub-cases we recover certain systems equivalent to known Painlevé equations, or variants thereof. The resulting picture is a multi-faceted description of each case: the local behaviour around singularities, the surface type, and the Newton polygon.

Figures (18)

• Figure 1: Cascades of blow-up transformations for the Hamiltonian systems related to (from left to right) $H_{\text{C}1}$, $H_{\text{C}2}$, $H_{\text{C}3}$.
• Figure 2: Rational surface constructed for $H_{\text{C}1}$ and its corresponding intersection diagram (left), coinciding with the Dynkin diagram $E_6^{(1)}$ for $\text{P}_{\text{IV}}$. On the right the same for $H_{\text{C}2}$, whose intersection diagram coincides with $E_7^{(1)}$ for $\text{P}_{\text{II}}$.
• Figure 3: Rational surface constructed for $H_{\text{C}3}$ and its corresponding intersection diagram (left), coinciding with the Dynkin diagram $E_8^{(1)}$ for $\text{P}_{\text{I}}$.
• Figure 4: Cascades of blow-up transformations for the Hamiltonian systems related to (from left to right) $H_{\text{Q}1}$ in section \ref{['sec:4_simple_points']}, $H_{\text{Q}2}$ in section \ref{['sec:1_double_2_simple']}, $H_{\text{Q}3}$ in section \ref{['sec:2_double_points']} and $H_{\text{Q}4}$ in section \ref{['sec:1_triple_1_simple_point']}.
• Figure 5: Rational surface constructed for $H_{\text{Q}1}$ (left) and its corresponding intersection diagram (right).

Theorems & Definitions (4)

• Remark 4.1
• Remark 5.1
• Remark 5.2
• Remark 5.3